Comm. Algebra 32(5) (2004) 2015-2017

Integrally closed domains, minimal polynomials, and null ideals of matrices

Abstract: We show that every element of the integral closure D' of a domain D occurs as a coefficient of the minimal polynomial of a matrix with entries in D. This answers affirmatively a question of J. Brewer and F. Richman, namely, if integrally closed domains are characterized by the property that the minimal polynomial of every square matrix with entries in D is in D[x]. It follows that a domain D is integrally closed if and only if for every matrix A with entries in D the null ideal of A (consisting of all polynomials f in D[x] with f(A)=0) is a principal ideal of D[x].

2000 Mathematics Subject Classification: Primary 13B22, 15A21; Secondary 13B25, 12E05, 11C08, 11C20

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